I
A GRAVITATIONAL WAVE IN A MICHELSON INTERFEROMETER
Figure 1: Simple Michelson interferometer: a laser with power PL shoots light onto a beam
splitter with reflection and transmission coeffiecients rbs and tbs
A simple Michelson interferometer uses a laser producing planar monochromatic light waves
having a total power PL . The reflection of a 50-50 beam splitter can be modelled multiplying
p
the incoming electric field component amplitude
of
the
lightwave
by
a
factor
r
=
1/
2
bs
p
for reflection from one side and rbsp= 1/ 2 from the other side, while the transmission
multiplies it always with tb s = 1/ 2. Mirrors multiply that amplitude by r1 and r2 for
mirror 1 and mirror 2, respectively.
(I.1) What is the power output Pout,static of a Michelson with a static arm length di↵erence
Lstatic = L2 L1 ?
(I.2) Define 0 = 4⇡/ ⇥ Lstatic as the static phase di↵erence between between the two
returning beams at the beam splitter. What is the (time varying) phase di↵erence on the
beams introduced by a passing gravitational wave, optimally oriented and with the ’+’polarization? Introduce this
GW (t) in the expression found at (I.1) and explain why you
chose to put it there.
(I.3) Use the answer found at (I.2) and the trigonomtic identity cos(a + b) = cos(a)cos(b)
sin(a)sin(b) to extract the static part (Pout,static , same as found at (I.1)) and a time varying
part Pout,GW (t). At what value for 0 is Pout,GW (t) maximized?
II
MAXIMIZING SIGNAL-TO-NOISE RATIO (SNR)
In the absence of noise, the condition determined at (I.3) would be the most sensitive condition of the interferometer. However, the quantum nature of light results in a statistical
fluctuation in the arrival times of the photons on the photo-diode. This is called shot noise
and brings an uncertainty to the intensity measurement in which a GW signal could be hiding. To really maximize the sensitivity of the detector, the signal-to-noise ratio (SNR) needs
1
to be maximized.
(II.1) If SGW is the amplitude spectral density (ASD) of the phase induced by the gravitational
wave, what is the ASD of the signal?
p
(II.2) Define the ASD of the shot noise is Sshot = 2Pout,static ~!, with ! the angular frequency
of the laser light. Show that
r
PL r12 r22
SN R =
f ( )SGW ,
(1)
2~!
where f ( ) = sin2
2
0 /(r1
+ r22
2r1 r2 cos 0 ).
(II.3) At the dark fringe, i.e. 0,df = 2n⇡, to what order is the signal in h? Is operating a
gravitational wave detector in this condition the best idea? Could you also draw a similar
conclusion from looking at eq. (1) when you assume the r1 ⇡= r2 ⇡ 1?
(II.4) The SNR is optimised when f( ) is maximal. Show that maximum is not exactly at the
dark fringe, i.e. cos( 00 ) = r1 /r2 . 1, with 00 (a phase close to the dark fringe) if you assume
the reflectivities to be slightly smaller than 1 and one slightly di↵erent from the other (choose
r1 < r2 ). Then, assuming r1 ⇡ r2 ⇡ 1, show that 00 maximizes the SNR with f ( 00 ) ⇡ 1.
III
THE SENSITIVITY OF A SHOT NOISE LIMITED INTERFEROMETER
The sensitivity of an interferometer is defined as the ASD of the strain for which SNR=1.
The goal of this exercise is to compute the sensitivity of the Michelson interferometer studied
in the previous exercise parts.
(III.1) What is the ASD for the phase of a gravitational wave with SNR = 1?
(III.2) What is the ASD for the strain of the gravitational wave (with SNR=1)? (Hint:
compute first the phase di↵erence at the beam splitter assuming that the travel time of the
photon in the interferometer arms is much smaller then the period of the gravitational wave.
How do the ASD of the phase and the strain of the gravitational wave relate?)
(III.3) Assume an arm length of 3 km, an input laser power of 4 W, and a wavelength of
1064 nm. What is the value of the strain in the previous exercise?
(III.4) At (III.2) you found that the strain of a shot noise limited interferometer does not depend on the gravitational wave frequency. What does this imply for the bandwidth/sources?
(Hint: the typical gravitational wave frequency of a source is source dependent: binary neutron stars emit typically gravitational waves of higher frequencies during merger than binary
black holes)
(III.5) The sensitivity of the actual detectors is frequency dependent - what is unrealistic in
the former computation?
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